Finally 2 is also closely related to a quantity

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Unformatted text preview: of the edge S , where · is a matrix norm, and 1 is a vector of ones of length |S |. (a) Show that the optimization problem minimize edges S fS ( X ) is convex in the free node coordinates xj . (b) The size fS (X ) of a net S obviously depends on the norm used in the definition (41). We consider five norms. • Frobenius norm: Xs − y 1T F = j ∈S i=1 • Maximum Euclidean column norm: 1/2 p (xij − yi )2 1/2 p X S − y 1T = max 2,1 j ∈S i=1 • Maximum column sum norm: (xij − yi )2 p X S − y 1T 1,1 = max j ∈S i=1 |xij − yi |. • Sum of absolute values norm: p X s − y 1T sav 127 . = j ∈S i=1 |xij − yi | . • Sum-row-max norm: p X s − y 1T srm = i=1 max |xij − yi | j ∈S For which of these norms does fS have the following interpretations? (i) fS (X ) is the radius of the smallest Euclidean ball that contains the nodes of S . (ii) fS (X ) is (proportional to) the perimeter of the smallest rectangle that contains the nodes of S : p 1 fS ( X ) = (max xij − min xij ). j ∈S 4 i=1 j ∈S (iii) fS (X ) is the squ...
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